**HAL Id: hal-02343861**

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**Opetopic algebras I: Algebraic structures on opetopic**

**sets**

### Cédric Ho Thanh, Chaitanya Leena Subramaniam

**To cite this version:**

Cédric Ho Thanh, Chaitanya Leena Subramaniam. Opetopic algebras I: Algebraic structures on opetopic sets. 2019. �hal-02343861v2�

CÉDRIC HO THANH AND CHAITANYA LEENA SUBRAMANIAM

Abstract. We define a family of structures called “opetopic algebras”, which are algebraic structures with an underlying opetopic set. Examples of such are categories, planar operads, and Loday’s combinads over planar trees. Opetopic algebras can be defined in two ways, either as the algebras of a “free pasting diagram” parametric right adjoint monad, or as models of a small projective sketch over the category of opetopes. We define an opetopic nerve functor that fully embeds each category of opetopic algebras into the category of opetopic sets. In particular, we obtain fully faithful opetopic nerve functors for categories and for planar colouredSet-operads.

This paper is the first in a series aimed at using opetopic spaces as models for higher algebraic structures. Contents 1. Introduction 2 1.1. Context 2 1.2. Contributions 3 1.3. Outline 3 1.4. Related work 3 1.5. Acknowledgments 3

1.6. Preliminary category theory 4

2. Polynomial functors and polynomial monads 5

2.1. Polynomial functors 5

2.2. Trees 6

2.3. Addresses 7

2.4. Grafting 8

2.5. Polynomial monads 9

2.6. The Baez–Dolan construction 9

3. Opetopes 10

3.1. Polynomial definition of opetopes 11

3.2. Higher addresses 11

3.3. The category of opetopes 12

3.4. Opetopic sets 13

3.5. Extensions 16

4. The opetopic nerve of opetopic algebras 16

4.1. Opetopic algebras 16

4.2. Parametric right adjoint monads 17

4.3. Coloured Z*n*_{-algebras} _{19}

4.4. Z*n*-cardinals and opetopic shapes 21

4.5. The opetopic nerve functor 25

5. Algebraic trompe-l’œil 29

5.1. Colour 29

5.2. Shape 30

*Date: October 2019.*

*2010 Mathematics Subject Classification. Primary 18C20; Secondary 18C30.*

*Key words and phrases. Opetope, Opetopic set, Operad, Polynomial monad, Projective sketch.*
1

Appendix A. Omitted proofs 33

Index 38

References 40

1. Introduction

*This paper deals with algebraic structures whose operations have higher dimensional “tree-like” *

*ar-ities. As an example in lieu of a definition, a category* _{C is an algebraic structure whose operation of}
*composition has as its inputs, or “arities”, sequences of composable morphisms of the category. These*

*sequences can can be seen as filiform or linear trees. Moreover, each morphism of*_{C can itself be seen as}
an operation whose arity is a single point (i.e. an object ofC). A second example, one dimension above,
is that of a planar coloured _{Set-operad P (a.k.a. a nonsymmetric multicategory), whose operation of}
composition has planar trees of operations (a.k.a. multimorphisms) of _{P as arities. Moreover, the arity}
of an operation ofP is an ordered list (a filiform tree) of colours (a.k.a. objects) of P. Heuristically
ex-tending this pattern leads one to presume that such an algebraic structure one dimension above planar
operads should have an operation of composition whose arities are trees of things that can themselves
be seen as operations whose arities are planar trees. Indeed, such algebraic structures are precisely the
*PT-combinads in Set (combinads over the combinatorial pattern of planar trees) of Loday [Lod12].*

Structure _{=0-algebras}Sets _{=1-algebras}Categories _{=2-algebras}Operads PT−combinads_{=3-algebras} ⋯
Arity _{=1-opetopes}{∗} _{=2-opetopes}Lists _{=3-opetopes}Trees 4-opetopes ⋯

The goal of this article is to give a precise definition of the previous sequence of algebraic structures.
**1.1. Context. It is well-known that higher dimensional tree-like arities are encoded by the opetopes**
*(operation polytopes) of Baez and Dolan [BD98], which were originally introduced in order to give a*
*definition of weak n-categories and a precise formulation of the “microcosm” principle.*

The fundamental definitions of [BD98] are those of * _{P-opetopic sets and n-coherent P-algebras (for}*
a coloured symmetric

_{Set-operad P), the latter being P-opetopic sets along with certain “horn-filling”}operations that are “universal” in a suitable sense. When

_{P is the identity monad on Set (i.e. the }uni-colour symmetric

*Set-operad with a single unary operation), P-opetopic sets are simply called opetopic*

*sets, and n-coherent _{P-algebras are the authors’ proposed definition of weak n-categories.}*

While the coinductive definitions in [BD98] of*P-opetopic sets and n-coherent P-algebras are *
*straight-forward and general, they have the disadvantage of not defining a category of* _{P-opetopes such that}
presheaves over it are precisely _{P-opetopic sets, even though this is ostensibly the case. Directly }
defin-ing the category of _{P-opetopes turns out to be a tedious and non-trivial task, and was worked out}
explicitly by Cheng in [Che03, Che04a] for the particular case of the identity monad on _{Set, giving the}
category O of opetopes.

The complexity in the definition of a category _{O}_{P} of _{P-opetopes has its origin in the difficulty of}
*working with a suitable notion of symmetric tree. Indeed, the objects of* O_{P} are trees of trees of ... of
trees of operations of the symmetric operad_{P, and their automorphism groups are determined by the}
action of the (“coloured”) symmetric groups on the sets of operations of_{P.}

However, when the action of the symmetric groups on the sets of operations of _{P is free, it turns}
out that the objects of the category _{O}_{P} are rigid, i.e. have no non-trivial automorphisms (this follows
from [Che04a, proposition 3.2]). The identity monad onSet is of course such an operad, and this vastly
simplifies the definition of* _{O. Indeed, such operads are precisely the (finitary) polynomial monads in Set,}*
and the machinery of polynomial endofunctors and polynomial monads developed in [Koc11, KJBM10,
GK13] gives a very satisfactory definition of

_{O [HT18, CHTM19] which we review in sections 2 and 3.}

**1.2. Contributions. The main contribution of the present article is to show how the polynomial **
defi-nition of* _{O allows, for all k, n ∈ N with k ≤ n, a definition of (k, n)-opetopic algebras, which constitute a}*
full subcategory of the category

_{Psh(O) of opetopic sets. More precisely, we show that the polynomial}monad whose set of operations is the set O

*n*+1 of

*(n + 1)-dimensional opetopes can be extended to a*

parametric right adjoint monad whose algebras are the * _{(k, n)-opetopic algebras. Important particular}*
cases are the categories of

*(1, 1)- and (1, 2)-opetopic algebras, which are the categories Cat and Op*colof

small categories and coloured planar_{Set-operads respectively. Loday’s combinads over the combinatorial}
pattern of planar trees [Lod12] are also recovered as*(1, 3)-opetopic algebras.*

We further show that each category of_{(k, n)-opetopic algebras admits a fully faithful opetopic nerve}

*functor to* _{Psh(O). As a direct consequence of this framework, we obtain commutative triangles of}

adjunctions
Cat
Psh(O) **Psh(∆)**
*N*
⊥
*Nu*
⊥
*h*
*h*!
*u*
*h*∗
⊥
Opcol
Psh(O≥1) **Psh(Ω),***N*
⊥
*Nu*
⊥
*h*
*h*!
*u*
*h*∗
⊥
(1.2.1)

* where ∆ is the category of simplices, Ω is the planar version of Moerdijk and Weiss’s category of dendrices*
andO

_{≥1}is the full subcategory ofO on opetopes of dimension > 0. This gives a direct comparison between the opetopic nerve of a category (resp. a planar operad) and its corresponding well-known simplicial (resp. dendroidal) nerve.

This formalism seems to provide infinitely many types of* _{(k, n)-opetopic algebras. However this is not}*
really the case, as the notion stabilises at the level of combinads. Specifically, we show a phenomenon

*we call algebraic trompe-l’œil, where an*

*(k, n)-opetopic algebra is entirely specified by its underlying*opetopic set and by a

*into a*

_{(1, 3)-opetopic algebra. In other words, its algebraic data can be “compressed”}*(1, 3)-algebra (a combinad). The intuition behind this is that fundamentally, opetopes are just*trees whose nodes are themselves trees, and that once this is obtained at the level of combinads, opetopic algebras can encode no further useful information.

**1.3. Outline. We begin by recalling elements of the theory of polynomial functors and polynomial**
monads in section 2. This formalism is the basis for the modern definition of opetopes and of the
category of opetopes [KJBM10, CHTM19] that we survey in section 3. Section 4 contains the central
constructions of this article, namely those of opetopic algebras and coloured opetopic algebras, as well
as the definition of the opetopic nerve functor, which is a full embedding of (coloured) opetopic algebras
into opetopic sets. Section 5 is devoted to showing how the algebraic information carried by opetopes
turns out to be limited.

**1.4. Related work. The** _{(k, n)-opetopic algebras that we obtain are related to the n-coherent }*P-algebras of [BD98] as follows: for n≥ 1, (1, n)-opetopic algebras are precisely 1-coherent P-algebras for*
P the polynomial monad O*n*−1 *←Ð En* Ð→ O*n* Ð→ O*n*−1*. We therefore do not obtain all n-coherent*

*P-algebras with our framework, and this means in particular that we cannot capture all the weak *
*n-categories of [BD98] (except for n* _{= 1, which are just usual 1-categories). This is not unexpected, as}
*weak n-categories are not defined just by equations on the opetopes of an opetopic set, but by its more*
*subtle universal opetopes.*

However, we are able to promote the triangles of (1.2.1) to Quillen equivalences of simplicial model
categories. This, along with a proof that opetopic spaces (i.e. simplicial presheaves on*O) model (∞, *
1)-categories and planar_{∞-operads, will be the subject of the second paper of this series.}

**1.5. Acknowledgments. We are grateful to Pierre-Louis Curien for his patient guidance and reviews.**
The second named author would like to thank Paul-André Melliès for introducing him to nerve functors

and monads with arities, Mathieu Anel for discussions on Gabriel–Ulmer localisations, and Eric Finster for a discussion related to proposition 2.6.5. The first named author has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska–Curie grant agreement No. 665850.

**1.6. Preliminary category theory. We review relevant notions and basic results from the theory of**
locally presentable categories. Original references are [GU71, SGA72], and most of this material can be
found in [AR94].

*1.6.1. Presheaves and nerve functors. For* _{C ∈ Cat, we write Psh(C) for the category of Set-valued}
presheaves over _{C, i.e. the category of functors C}op_{Ð→ Set and natural transformations between them.}

*If X* *∈ Psh(C) is a presheaf, then its category of elements C/X is the comma category y/X, where*
y∶ C ↪Ð→ Psh(C) is the Yoneda embedding.

Let *C ∈ Cat and F ∶ C Ð→ D be a functor to a (not necessarily small) category D. Then the nerve*

*functor associated to F (also called the nerve of F ) is the functor NF* *∶ D Ð→ Psh(C) mapping d ∈ D to*

*D(F −, d). The functor F is said to be dense if for all d ∈ D, the colimit of F /d Ð→ C*Ð→ D exists in D,*F*
*and is canonically isomorphic to d. Equivalently, F is dense if and only if N _{F}* is fully faithful.

*Let i∶ C Ð→ D be a functor between small categories. Then the precomposition functor i*∗∶ Psh(D) Ð→
*Psh(C) has a left adjoint i*! *and a right adjoint i*∗*, given by left and right Kan extension along i*op

*respectively. If i has a right adjoint j, then i*∗ * _{⊣ j}*∗

*, or equivalently, i*∗

_{≅ j}_{!}

*. Note that the nerve of i is*

*the functor Ni= i*∗yD, where yD∶ D ↪

*Ð→ Psh(D) is the Yoneda embedding. Recall that i*∗ is the nerve of

the functor y_{D}*i and that i*_{∗} *is the nerve of the functor Ni* *= i*∗y_{D}*, i.e. it is the nerve of the nerve of i.*

*1.6.2. Orthogonality. Let _{C be a category, and l, r ∈ C}*→

*. We say that l is left orthogonal to r (equivalently,*

*r is right orthogonal to l), written l⊥ r, if for any solid commutative square as follows, there exists a*
*unique dashed arrow making the two triangles commute (the relation* _{⊥ is also known as the unique}*lifting property):*

⋅ ⋅

⋅ ⋅

*l* *r*

If _{C has a terminal object 1, then for all X ∈ C, we write l ⊥ X if l is left orthogonal to the unique}*map X _{Ð→ 1. Let L and R be two classes of morphisms of C. We write L ⊥ R if for all l ∈ L and r ∈ R we}*

*have l⊥ r. The class of all morphisms f such that L ⊥ f (resp. f ⊥ R) is denoted L*⊥ (resp⊥R).

*1.6.3. Localisations. Let J be a class of morphisms of a category _{C. Recall that the localisation of C at J}*

*is a functor γ*

_{J}

_{∶ C Ð→ J}−1

_{C such that γ}_{J}

*f is an isomorphism for every f*

*∈ J, and such that γ*J is universal

*for this property. We say that J has the 3-for-2 property when for every composable pair* _{⋅}_{Ð}*f*_{→ ⋅}_{Ð}_{→ ⋅ of}*g*
morphisms in*C, if any two of f, g and gf are in J, then so is the third.*

Assume now that* _{C is a small category, and that J is a set (rather than a proper class) of morphisms}*
ofPsh(C), and consider the full subcategory CJ↪

*Ð→ Psh(C) of all those X ∈ Psh(C) such that J ⊥ X. A*

*category is locally presentable if and only if it is equivalent to one of the form*_{C}_{J}. The pair _{(}⊥_{(J}⊥* _{), J}*⊥

_{)}

*forms an orthogonal factorisation system, meaning that any morphism f in*Psh(C) can be factored as

*f* *= pi, where p ∈ J*⊥ *and i*_{∈} ⊥_{(J}⊥_{). Applied to the unique arrow X Ð→ 1, this factorisation provides a}*left adjoint (i.e. a reflection) a*_{J}_{∶ Psh(C) Ð→ C}_{J} to the inclusion_{C}_{J} _{↪}_{Ð→ Psh(C). Furthermore, a}_{J} is the
localisation ofPsh(C) at J. 1

With _{C and J as in the previous paragraph, the class of J-local isomorphisms W}_{J} is the class of
*all morphisms f* ∈ Psh(C)→ *such that for all X* ∈ CJ*, f* *⊥ X (that is, W*J = ⊥CJ). It is the smallest
1_{The results of this paragraph still hold when}_{Psh(C) is replaced by a locally κ-presentable category E and a set K of}*κ-small morphisms ofE. We call a localisation of the form a*K∶ E Ð→ EK*the Gabriel–Ulmer localisation of*E at K.

class of morphisms that contains J, that satisfies the 3-for-2 property, and that is closed under colimits
in _{Psh(C)}→ *[GU71, theorem 8.5]. Thus the localisation a*_{J} is also the localisation of _{Psh(C) at W}_{J}.
Furthermore, W_{J} is closed under pushouts.

*1.6.4. Projective sketches. A projective sketch is the data of a* *C ∈ Cat and a set K of cones in C.*
For _{B a category with all limits, the category of models in B of (C, K) is the category of functors}*C Ð→ B that take each cone in K to a limit cone, and natural transformations between them. If (C, K)*
*is a projective sketch, then equivalently, K can be seen as a set of cocones in* Cop_{, or as a set of }

sub-representables (subobjects of sub-representables) in_{Psh(C}op_{). Let C}op

*K* ↪Ð→ Psh(C

op_{) be the full subcategory}

*of all X*_{∈ Psh(C}op* _{) such that K ⊥ X. Then C}*op

*K* is precisely the category of models inSet of the projective

sketch*(C, K). Let κ be a regular cardinal, and let (C, K) be a projective sketch in which each cone in K*
*is a κ-small diagram. Then* _{C}op_{K}*is a locally κ-presentable category, and the inclusion*_{C}op_{K}_{↪}_{Ð→ Psh(C}op_{)}

*preserves κ-filtered colimits.*

2. Polynomial functors and polynomial monads

We survey elements of the theory of polynomial functors, trees, and monads. For more comprehensive references, see [Koc11, GK13].

**2.1. Polynomial functors.**

* Definition 2.1.1 (Polynomial functor). A polynomial (endo)functor P over I is a diagram in* Set of

the form

*I* *s* *E* *p* *B* *t* *I.* (2.1.2)

*P is said to be finitary if the fibres of p∶ E Ð→ B are finite sets. We will always assume polynomial*

functors to be finitary.

*We use the following terminology for a polynomial functor P as in equation (2.1.2), which is motivated*
by the intuition that a polynomial functor encodes a multi-sorted signature of function symbols. The
*elements of B are called the nodes or operations of P , and for every node b, the elements of the fibre*

*E(b) ∶= p*−1*(b) are called the inputs of b. The elements of I are called the colours or sorts of P . For every*

*input e of a node b, we denote its colour by s _{e}_{(b) ∶= s(e).}*

*b*

*se*1*b* ⋯ *sekb*

*t(b)*

*e*

1 _{e}k

**Definition 2.1.3 (Morphism of polynomial functor). A morphism from a polynomial functor P over**

*I (as in equation (2.1.2)) to a polynomial functor P*′ *over I*′ (on the second row) is a commutative
diagram of the form

*I* *E* *B* *I*
*I*′ *E*′ *B*′ *I*′
*f*0 ⌟
*p*
*f*2
*s* *t*
*f*1 *f*0
*p*′
*s*′ *t*

*where the middle square is cartesian (i.e. is a pullback square). If P and P*′ are both polynomial
*functors over I, then a morphism from P to P*′ *over I is a commutative diagram as above, but where f*0

is required to be the identity. Let _{PolyEnd denote the category of polynomial functors and morphisms}
of polynomial functors, and_{PolyEnd(I) the category of polynomial functors over I and morphisms of}*polynomial functors over I.*

*Remark 2.1.4 (Polynomial functors really are functors!). The term “polynomial (endo)functor” is due*

*to the association of P to the composite endofunctor*

*Set/I*↪*Ð→ Set/Es*∗ ↪*Ð→ Set/Bp*∗ *t*!

↪*Ð→ Set/I*

*where we have denoted a*_{!} *and a*_{∗} *the left and right adjoints to the pullback functor a*∗ along a map of
*sets a. Explicitly, for _{(X}_{i}_{∣ i ∈ I) ∈ Set/I, P (X) is given by the “polynomial”}*

*P(X) =*⎛
⎝*b∈B(j)*∑
∏
*e∈E(b)*
*X _{s}_{(e)}∣ j ∈ I*⎞
⎠

*,*(2.1.5)

*where B _{(i) ∶= t}*−1

*−1*

_{(i) and E(b) = p}

_{(b). Visually, elements of P (X)}_{j}*are nodes b*and whose inputs are decorated by elements of

_{∈ B such that tb = j,}*(Xi*

*∣ i ∈ I) in a manner compatible with their colours.*

*Graphically, an element of P X _{i}* can be represented as

*b*
*x*1 *xk*
⋯
*t(b)*
*e*
1 *ek*

*with b _{∈ B such that t(b) = i, and x}_{j}_{∈ X}_{s}_{ej}_{b}* for 1

_{≤ j ≤ k. Moreover, the endofunctor P ∶ Set/I Ð→ Set/I}*preserves connected limits: s*∗

*and p*

_{∗}

*preserve all limits (as right adjoints), and t*!preserves and reflects

connected limits.

This construction extends to a fully faithful functor *PolyEnd(I) Ð→ Cart(Set/I), the latter being*
the category of endofunctors of*Set/I and cartesian natural transformations*2. In fact, the image of this
full embedding consists precisely of those endofunctors that preserve connected limits [GK13, section
1.18]. The composition of endofunctors gives _{Cart(Set/I) the structure of a monoidal category, and}*PolyEnd(I) is stable under this monoidal product [GK13, proposition 1.12]. The identity polynomial*
*functor I* * _{←Ð I Ð→ I Ð→ I is associated to the identity endofunctor; thus PolyEnd(I) is a monoidal}*
subcategory of

*Cart(Set/I).*

**2.2. Trees.**

**Definition 2.2.1 (Polynomial tree). A polynomial functor T given by**

*T*0 *T*2 *T*1 *T*0

*s* *p* *t*

*is a (polynomial) tree [Koc11, section 1.0.3] if*

*(1) the sets T*_{0}*, T*_{1} *and T*_{2} are finite (in particular, each node has finitely many inputs);
*(2) the map t is injective;*

*(3) the map s is injective, and the complement of its image T*0*− im s has a single element, called*

*the root;*

*(4) let T*0 *= T*2 *+ {r}, with r the root, and define the walk-to-root function σ by σ(r) = r, and*

*otherwise σ _{(e) = tp(e); then we ask that for all x ∈ T}*0

*, there exists k∈ N such that σk(x) = r.*

*We call the colours of a tree its edges and the inputs of a node the input edges of that node.*

Let_{Tree be the full subcategory of PolyEnd whose objects are trees. Note that it is the category of}

*symmetric or non-planar trees (the automorphism group of a tree is in general non-trivial) and that its*

*morphisms correspond to inclusions of non-planar subtrees. An elementary tree is a tree with at most*
*one node. Let el*Tree be the full subcategory of Tree spanned by elementary trees.

**Definition 2.2.2 (P -tree). For P***∈ PolyEnd, the category tr P of P -trees is the slice Tree/P . The*

fundamental difference between* _{Tree and tr P is that the latter is always rigid i.e. it has no non-trivial}*
automorphisms [Koc11, proposition 1.2.3]. In particular, this implies that

_{PolyEnd does not have a}terminal object.

*Notation 2.2.3. Every P -tree T* _{∈ tr P corresponds to a morphism from a tree (which we shall denote by}

*⟨T ⟩) to P , so that T ∶ ⟨T ⟩ Ð→ P . We point out that ⟨T ⟩*1 is the set of nodes of*⟨T ⟩, while T*1*∶ ⟨T ⟩*1*Ð→ P*1

is a decoration of the nodes of*⟨T ⟩ by nodes of P , and likewise for edges.*

**Definition 2.2.4 (Category of elements). For P*** _{∈ PolyEnd, its category of elements}*3

_{elt P is the slice}*elTree/P . Explicitly, for P as in equation (2.1.2), the set of objects of elt P is I + B, and for each b ∈ B,*

there is a morphism t_{∶ t(b) Ð→ b, and a morphism s}_{e}_{∶ s}_{e}_{(b) Ð→ b for each e ∈ E(b). Remark that there}*is no non-trivial composition of arrows in elt P .*

**Proposition 2.2.5 ([Koc11, proposition 2.1.3]). There is an equivalence of categories***Psh(elt P ) ≃*

*PolyEnd/P .*

*Proof. For X _{∈ Psh(elt P ), construct the following polynomial functor over P :}*

∑*i∈IXi* *EX* ∑*b∈BXb* ∑*i∈IXi*

*I* *E* *B* *I,*

⌟

*where EX* Ð→ ∑*i∈IXi* *is given by the maps X*s*e* *∶ Xb* *Ð→ X*s*eb, for b* *∈ B and e ∈ E(b). In the other*

*direction, note that the full inclusion el*_{Tree ↪}* _{Ð→ PolyEnd induces a full inclusion ι ∶ elt P ↪}_{Ð→ PolyEnd/P}*
whose nerve functor

*PolyEnd/P Ð→ Psh(elt P ) maps Q ∈ PolyEnd/P to PolyEnd/P (ι−, Q). The two*constructions are easily seen to define the required equivalence of categories.

_{□}

**2.3. Addresses.**

**Definition 2.3.1 (Address). Let T**_{∈ Tree be a polynomial tree and σ be its walk-to-root function}

*(definition 2.2.1). We define the address function & on edges inductively as follows:*
*(1) if r is the root edge, let &r*∶=[],

*(2) if e _{∈ T}*0

*− {r} and if &σ(e) = [x], define &e ∶=[xe].*

*The address of a node b _{∈ T}*1

*is defined as &b∶= &t(b). Note that this function is injective since t is. Let*

*T*● *denote its image, the set of node addresses of T , and let T*∣ be the set of addresses of leaf edges, i.e.
*those not in the image of t.*

*Assume now that T* _{∶ ⟨T ⟩ Ð→ P is a P -tree. If b ∈ ⟨T ⟩}_{1} *has address &b _{= [p], write s}_{[p]}T∶= T*1

*(b). For*

*convenience, we let T*●* _{∶= ⟨T ⟩}*●

*, and T*∣

*∣.*

_{∶= ⟨T ⟩}*Remark 2.3.2. The formalism of addresses is a useful bookkeeping syntax for the operations of grafting*

and substitution on trees. The syntax of addresses will extend to the category of opetopes and will allow us to give a precise description of the composition of morphisms in the category of opetopes (see definition 3.3.2) as well as certain constructions on opetopic sets.

*Notation 2.3.3. We denote by tr*∣*P the set of P -trees with a marked leaf, i.e. endowed with the address*

of one of its leaves. Similarly, we denote by tr●*P the set of P -trees with a marked node.*

**2.4. Grafting.**

**Definition 2.4.1 (Elementary P -trees). Let P be a polynomial endofunctor as in equation (2.1.2). For**

*i∈ I, define Ii∈ tr P as having underlying tree*

*{i}* ∅ ∅ *{i},* (2.4.2)

*along with the obvious morphism to P , that which maps i to i∈ I. This corresponds to a tree with no*
*nodes and a unique edge decorated by i. Define Y _{b}_{∈ tr P , the corolla at b, as having underlying tree}*

*s(E(b)) + {∗}* *s* *E(b)* *{b}* *s(E(b)) + {∗},* (2.4.3)

*where the right map sends b to* ∗, and where the morphism Y*b* *Ð→ P is the identity on s(E(b)) ⊆ I,*

maps_{∗ to t(b) ∈ I, is the identity on E(b) ⊆ E, and maps b to b ∈ B. This corresponds to a P -tree with}*a unique node, decorated by b. Observe that for T* *∈ tr P , giving a morphism Ii* *Ð→ T is equivalent to*

specifying the address_{[p] of an edge of T decorated by i. Likewise, morphisms of the form Y}_{b}_{Ð→ T are}*in bijection with addresses of nodes of T decorated by b.*

**Definition 2.4.4 (Grafting). For S, T*** _{∈ tr P , [l] ∈ S}*∣

*such that the leaf of S at*

_{[l] and the root edge of}*T are decorated by the same i∈ I, define the grafting S ○ _{[l]}T of S and T on[l] by the following pushout*

*(in tr P ):*
I*i* *T*
*S* *S*○* _{[l]}T.*
⌜
[]

*[l]*(2.4.5)

*Note that if S (resp. T ) is a trivial tree, then S*○_{[l]}T*= T (resp. S). We assume, by convention, that the*

grafting operator_{○ associates to the right.}

**Proposition 2.4.6 ([Koc11, proposition 1.1.21]). Every P -tree is either of the form I**_{i}, for some i_{∈ I,}*or obtained by iterated graftings of corollas (i.e. P -trees of the form Yb* *for b∈ B).*

*Notation 2.4.7 (Total grafting). Take T, U*_{1}*, . . . , Uk* *∈ tr P , where T*∣ *= {[l*1*], . . . , [lk*]}, and assume the

*grafting T*○_{[l}_{i}_{]}*Ui* *is defined for all i. Then the total grafting will be denoted concisely by*

*T*◯
*[li*]
*Ui= (⋯(T ○*
*[l*1]
*U*1) ○
*[l*2]
*U*2⋯) ○
*[lk*]
*Uk.* (2.4.8)

It is easy to see that the result does not depend on the order in which the graftings are performed.

**Definition 2.4.9 (Substitution). Let** * _{[p] ∈ T}*●

*and b*

_{= s}

_{[p]}T . Then T can be decomposed as*T*

*= A ○*

*[p]*Y*b _{[e}*◯

_{i}_{]}

*Bi,*(2.4.10)

*where E _{(b) = {e}*

_{1}

*, . . . , ek}, and A, B*1

*, . . . , Bk∈ tr P . For U a P -tree with a bijection ℘ ∶ U*∣

*Ð→ E(b) over*

*I, we define the substitution T*◽_{[p]}U as*T* ◽

*[p]U∶= A ○[p]U*_{℘}◯−1_{e}_{i}

*Bi.* (2.4.11)

In other words, the node at address*[p] in T has been replaced by U, and the map ℘ provided “rewiring*
*instructions” to connect the leaves of U to the rest of T .*

**2.5. Polynomial monads.**

**Definition 2.5.1 (Polynomial monad). A polynomial monad over I is a monoid in**PolyEnd(I). Note

*that a polynomial monad over I is thus necessarily a cartesian monad on* *Set/I.*4 Let *PolyMnd(I) be*
the category of monoids in _{PolyEnd(I). That is, PolyMnd(I) is the category of polynomial monads}*over I and morphisms of polynomial functors over I that are also monad morphisms.*

**Definition 2.5.2 (**(−)⋆ **construction). Given a polynomial endofunctor P as in equation (2.1.2), we***define a new polynomial endofunctor P*⋆ as

*I* *s* tr∣*P* *p* *tr P* *t* *I* (2.5.3)

*where s maps a P -tree with a marked leaf to the decoration of that leaf, p forgets the marking, and t*
*maps a tree to the decoration of its root. Remark that for T* * _{∈ tr P we have p}*−1

*T= T*∣.

* Theorem 2.5.4 ([Koc11, section 1.2.7], [KJBM10, sections 2.7 to 2.9]). The polynomial functor P*⋆

*has a canonical structure of a polynomial monad. Furthermore, the functor* (−)⋆ *is left adjoint to the*
*forgetful functor* _{PolyMnd(I) Ð→ PolyEnd(I), and the adjunction is monadic.}

**Definition 2.5.5 (Readdressing function). We abuse notation slightly by letting**_{(−)}⋆ denote the
asso-ciated monad on *PolyEnd(I). Let M be a polynomial monad as on the left below. Buy theorem 2.5.4,*

*M is a*(−)⋆*-algebra, and we will write its structure map M*⋆_{Ð→ M as on the right:}

*I* *s* *E* *p* *B* *t* *I,*

*I* tr∣*M* *tr M* *I*

*I* *E* *B* *I.*

⌟

℘ t (2.5.6)

*where rT is the decoration of the root edge of a tree T* *∈ tr M. We call ℘T* *∶ T*∣ ≅*Ð→ E(rT ) the readdressing*

*function of T , and t T* * _{∈ B is called the target of T . If we think of an element b ∈ B as the corolla Y}_{b}*,
then the target map t “contracts” a tree to a corolla, and since the middle square is a pullback, the
number of leaves is preserved. The map℘

*T*

*establishes a coherent correspondence between the set T*∣ of

*leaf addresses of a tree T and the elements of E _{(T ).}*

**2.6. The Baez–Dolan construction.**

**Definition 2.6.1 (Baez–Dolan**_{(−)}+**construction). Let M be a polynomial monad as in equation (2.1.2),***and define its Baez–Dolan construction M*+to be

*B* s tr●*M* *p* *tr M* t *B* (2.6.2)

*where s maps an M -tree with a marked node to the label of that node, p forgets the marking, and t is*
*the target map. If T* * _{∈ tr M, remark that p}*−1

*T*

*= T*●

*is the set of node addresses of T . If*●, then s

_{[p] ∈ T}*[p] ∶= s*

_{[p]}T .* Theorem 2.6.3 ([KJBM10, section 3.2]). The polynomial functor M*+

*has a canonical structure of a*

*polynomial monad.*

*Remark 2.6.4. The*_{(−)}+ construction is an endofunctor on _{PolyMnd whose definition is motivated as}
*follows. If we begin with a polynomial monad M , then the colours of M*+*are the operations of M . The*
*operations of M*+*, along with their output colour, are given by the monad multiplication of M : they*
*are the relations of M , i.e. the reductions of trees of M to operations of M . The monad multiplication*

4_{We recall that a monad is cartesian if its endofunctor preserves pullbacks and its unit and multiplication are cartesian}

*on M*+ *is given as follows: the reduction of a tree of M*+ *to an operation of M*+ *(which is a tree of M )*
*is obtained by substituting trees of M into nodes of trees of M .*

*Let M be a finitary (i.e. the fibers of p below are finite, or equivalently, p*_{∗}*, and therefore M* _{= t}_{!}*p*_{∗}*s*∗,
preserves filtered colimits) polynomial monad whose underlying polynomial functor is

*I* *s* *E* *p* *B* *t* *I.*

*The Baez–Dolan construction gives the polynomial monad M*+whose underlying polynomial functor is

*B* s tr●*M* *p* *tr M* t *B.*

Recall also from theorem 2.5.4 that the category *PolyMnd(I) is the category of (−)*⋆-algebras. The
following fact is analogous to proposition 2.2.5 and is at the heart of the Baez–Dolan construction
*(indeed, it is even the original definition of the construction, see [BD98, definition 15]).*

**Proposition 2.6.5. For M a polynomial monad, there is an equivalence of categories***Alg(M*+) ≃

*PolyMnd(I)/M.*

*Proof. Given a M*+*-algebra M*+*X*Ð*→ X in Set/B, define ΦX ∈ PolyEnd(I)/M asx*

*I* *EX* *X* *I*

*I* *E* *B* *I.*

⌟

*There is an evident bijection tr ΦX≅ M*+*X inSet/I, and the structure map x extends by pullback along*
*EX* *Ð→ X to a map (ΦX)*⋆ *Ð→ ΦX in PolyEnd(I). It is easy to verify that this determines a (−)*⋆

*-algebra structure on ΦX, and that the map ΦXÐ→ M in PolyEnd(I) is a morphism of (−)*⋆-algebras.
*Conversely, given an N _{∈ PolyMnd(I)/M whose underlying polynomial functor is}*

*I* *E*′ *B*′ *I,*

*then the bijection tr N* * _{≅ M}*+

*B*′ in

*⋆*

_{Set/I and the (−)}*-algebra map N*⋆

*+*

_{Ð→ N provide a map M}*B′ ΨN*ÐÐ→

*B*′in

*+-algebra and that the constructions*

_{Set/I. It is easy to verify that ΨN is the structure map of a M}Φ and Ψ are functorial and mutually inverse. □

*Remark 2.6.6. The previous result provides an equivalence between* _{PolyMnd(I)/M and the category}

*of M*+-algebras. A “coloured” version of this result can be (informally) stated as follows: forAlgcol* _{(M}*+

_{)}

*a suitable category of coloured M*+-algebras, there is an equivalence_{Alg}col_{(M}_{+}_{) ≃ PolyMnd/M, where}

*PolyMnd is the category of all polynomial monads (for all I in Set).*
3. Opetopes

In this section, we use the formalism of polynomial functors and polynomial monads of section 2 to
define opetopes and morphisms between them. This gives us a category_{O of opetopes and a category}
Psh(O) of opetopic sets. Our construction of opetopes is precisely that of [KJBM10], and by [KJBM10,
theorem 3.16], also that of [Lei04, Lei98], and by [Che04b, corollary 2.6], also that of [Che03]. As we
will see, the category*O is rigid, i.e. it has no non-trivial automorphisms (it is in fact a direct category).*

**3.1. Polynomial definition of opetopes.**

**Definition 3.1.1 (The Z***n* ** _{monad). Let Z}**0

_{be the identity polynomial monad on}

_{Set, as depicted on}

the left below, and let Z*n*_{∶=(Z}*n*−1_{)}+_{. Write Z}*n* _{as on right:}

{∗} {∗} {∗} *{∗},* O*n* *En*+1 O*n*+1 O*n.*

s *p* t _{(3.1.2)}

**Definition 3.1.3 (Opetope). An n-dimensional opetope (or simply n-opetope) ω is by definition an**

element of _{O}*n, and we write dim ω* *= n. An opetope ω ∈ On* *with n* *≥ 2 is called degenerate if its*

*underlying tree has no nodes (thus consists of a unique edge); it is non degenerate otherwise.*

*Following (2.5.6), for ω* ∈ O*n*+2, the structure of polynomial monad (Z*n*)⋆ Ð→ Z*n* gives a bijection

℘*ω∶ ω*∣*Ð→ (t ω)*● *between the leaves of ω and the nodes of t ω, preserving the decoration by n-opetopes.*
**Example 3.1.4.** (1) The unique 0-opetope is denoted *⧫ and called the point.*

(2) The unique 1-opetope is denoted _{◾ and called the arrow.}

*(3) If n* *≥ 2, then ω ∈ On* is a Z*n*−2-tree, i.e. a tree whose nodes are labeled in *(n − 1)-opetopes,*

and edges are labeled in* _{(n−2)-opetopes. In particular, 2-opetopes are Z}*0

_{-trees, i.e. linear trees,}

and thus in bijection with **N. We will refer to them as opetopic integers, and write n for the***2-opetope having exactly n nodes.*

**3.2. Higher addresses. By definition, an opetope ω of dimension n**_{≥ 2 is a Z}*n*−2_{-tree, and thus the}

*formalism of tree addresses (definition 2.3.1) can be applied to designate nodes of ω, also called its source*

*faces or simply sources. In this section, we iterate this formalism into the concept of higher dimensional*
*address, which turns out to be more convenient. This material is largely taken from [CHTM19, section*

2.2.3] and [HT18, section 4].

**Definition 3.2.1 (Higher address). Start by defining the set**_{A}_{n}*of n-addresses as follows:*
A0*= {∗} ,* A*n*+1= lists A*n,*

*where lists X is the set of finite lists (or words) on the alphabet X.*

Explicitly, the unique 0-address is _{∗ (also written [] by convention), while an (n + 1)-address is a}*sequence of n-addresses. Such sequences are enclosed by brackets. Note that the address* _{[], associated}
to the empty word, is in _{A}*n* *for all n*≥ 0. However, the surrounding context will almost always make

the notation unambiguous.

Here are examples of higher addresses:

[] ∈ A1*,* [∗ ∗ ∗ ∗] ∈ A1*,* [[][∗][]] ∈ A2*,* [[[[∗]]]] ∈ A4*.*

*For ω _{∈ O an opetope, nodes of ω can be specified uniquely using higher addresses, as we now show.}*

*Recall that En*−1 is the set of arities of Z

*n*−2(equation (3.1.2)). In Z0

*, set E*1(◾) = {∗}, so that the unique

node address of _{◾ is ∗ ∈ A}0*. For n* *≥ 2, recall that an opetope ω ∈ On* is a Z*n*−2*-tree ω* *∶ ⟨ω⟩ Ð→ Zn*−2

(definition 2.2.2), and write *⟨ω⟩ as*

*Iω* *Eω* *Bω* *Iω.*

*A node b* _{∈ B}ω*has an address &b* *∈ lists Eω, which by ω*2 *∶ Eω* *Ð→ En*−1 is mapped to an element

*of lists En*−1*. By induction, elements of En*−1 are *(n − 2)-addresses, whence ω*2*(&b) ∈ list An*−2 = A*n*−1.

*For the induction step, elements of En(ω) are nodes of ⟨ω⟩, which we identify by their aforementioned*

*(n − 1)-addresses. Consequently, for all n ≥ 1 and ω ∈ On, elements of En(ω) = ω*● can be seen as the set

of * _{(n − 1)-addresses of the nodes of ω, and similarly, ω}*∣ can be seen as the set of

_{(n − 1)-addresses of}*edges of ω.*

*We now identify the nodes of ω*_{∈ O}*n*+2 with their addresses. In particular, for*[p] ∈ ω*● a node address

which is an* _{(n + 1)-opetope. Let [l] = [p[q]] ∈ A}n*−1 be an address such that

*[p] ∈ ω*● and

*[q] ∈ (s*)●.

_{[p]}ωThen as a shorthand, we write

e* _{[l]}ω*∶= s

*s*

_{[q]}*(3.2.2)*

_{[p]}ω.**Example 3.2.3. Consider the 2-opetope on the left, called 3:**

. . . . ⇓ ◾ ◾ ◾ ⧫ ⧫ ⧫ ⧫ ∗ ∗ ∗ [] [∗] [∗∗]

Its underlying pasting diagram consists of 3 arrows ◾ grafted linearly. Since the only node address of
◾ is ∗ ∈ A0**, the underlying tree of 3 can be depicted as on the right. On the left of that three are the**

decorations: nodes are decorated with ◾ ∈ O1, while the edges are decorated with⧫ ∈ O0. For each node

in the tree, the set of input edges of that node is in bijective correspondence with the node addresses of
the decorating opetope, and that address is written on the right of each edges. In this low dimensional
example, those addresses can only be_{∗. Finally, on the right of each node of the tree is its 1-address,}
which is just a sequence of 0-addresses giving “walking instructions” to get from the root to that node.

**The 2-opetope 3 can then be seen as a corolla in some 3-opetope as follows:**

**3**
◾
◾ ◾ ◾
[]
[∗] [∗∗]
[]

As previously mentioned, the set of input edges is in bijective correspondence with the set of node
**addresses of 3.Here is now an example of a 3-opetope, with its annotated underlying tree on the right**
**(the 2-opetopes 1 and 2 are analogous to 3):**

*.*
*.* *.*
*.*
*.*
⇓ _{⇓}
⇓
⇛
*.*
*.* *.*
*.*
*.*
⇓
**3**
**1**
**2**
◾
◾
◾
◾
◾
◾ ◾
[]
[[∗∗]]
[[∗]]
[∗∗]
[∗]
[]
[]
[]
[∗]

**3.3. The category of opetopes. In this subsection, we define the category** _{O of planar opetopes}
introduced in [HT18], following the work of [Che03].

**Proposition 3.3.1 (Opetopic identities, [HT18, theorem 4.1]). Let ω**_{∈ O}*n* *with n≥ 2.*

*(1) (Inner edge) For* *[p[q]] ∈ ω*● *(forcing ω to be non degenerate), we have t s _{[p[q]]}ω*= s

*s*

_{[q]}

_{[p]}ω.*(2) (Globularity 1) If ω is non degenerate, we have t s*

_{[]}

*ω= t t ω.*

*(3) (Globularity 2) If ω is non degenerate, and* * _{[p[q]] ∈ ω}*∣

*, we have s*s

_{[q]}*= s*

_{[p]}ω_{℘}

_{ω}_{[p[q]]}t ω.*(4) (Degeneracy) If ω is degenerate, we have s*_{[]}*t ω= t t ω.*

**Definition 3.3.2 ([HT18, section 4.2]). With these identities in mind, we define the category** _{O of}

(1) (Objects) We set obO = ∑_{n}_{∈N}O*n*.

*(2) (Generating morphisms) Let ω* _{∈ O}*n* *with n* *≥ 1. We introduce a generator t ω*

t

Ð*→ ω, called*
*the target embedding. If* * _{[p] ∈ ω}*●, then we introduce a generator s

_{[p]}ω*ÐÐ→ ω, called a source*s

*[p]*

*embedding. A face embedding is either a source or the target embedding.*

(3) (Relations) We impose 4 relations described by the following commutative squares, that are well
*defined thanks to proposition 3.3.1. Let ω*∈ O*n* *with n*≥ 2

**(a) (Inner) for** * _{[p[q]] ∈ ω}*●

*(forcing ω to be non degenerate), the following square must*com-mute: s

*s*

_{[q]}*s*

_{[p]}ω*s*

_{[p]}ω

_{[p[q]]}ω*ω*s

*t s*

_{[q]}*[p]*s

_{[p[q]]}**(b) (Glob1) if ω is non degenerate, the following square must commute:**

*t t ω* *t ω*

s_{[]}*ω* *ω.*

t

t t

s_{[]}

* (c) (Glob2) if ω is non degenerate, and for[p[q]] ∈ ω*∣, the following square must commute:

s_{℘}_{ω}_{[p[q]]}t ω*t ω*

s_{[p]}ω*ω.*

s_{℘ω[p[q]]}

s_{[q]}_{t}

s_{[p]}

**(d) (Degen) if ω is degenerate, the following square must commute:**

*t t ω* *t ω*

*t ω* *ω.*

t

s_{[]} _{t}

t

See [HT18] for a graphical explanation of those relations.

*Notation 3.3.3. For n*∈ N, we let O* _{≤n}*be the full subcategory ofO spanned by opetopes of dimension at

*most n. The subcategories*

_{O}

*,*

_{<n}_{O}

*,*

_{≥n}_{O}

*, and*

_{>n}_{O}

*are defined similarly. Note that the latter is simply the set O*

_{=n}*n*.

**3.4. Opetopic sets. Recall from section 1.6 that**_{Psh(O) is the category of opetopic sets, i.e. Set-valued}
presheaves over_{O. For X ∈ Psh(O) and ω ∈ O, we will refer to the elements of the set X}_{ω}*as the cells*
*of X of shapeshape ω.*

**Definition 3.4.1.** *(1) The representable presheaf at ω*_{∈ O}* _{n}is denoted O_{[ω]. Its cells are morphisms}*
of

*O of the form f ∶ ψ Ð→ ω, for f a sequence of face embeddings, which we write fω ∈ O[ω]ψ*for

*short. For instance, the cell of maximal dimension is simply ω (as the corresponding sequence*
of face embeddings is empty), its * _{(n − 1)-cells are {s}_{[p]}ω∣ [p] ∈ ω*●

*} ∪ {t ω}, and there is no cell*

of dimension*> n.*

*(2) The boundary ∂O _{[ω] of ω is the maximal subpresheaf of O[ω] not containing the cell ω. We}*
write b

*ω∶ ∂O[ω] ↪Ð→ O[ω] for the boundary inclusion. The set of boundary inclusions is denoted*

*(3) The spine S[ω] is the maximal subpresheaf of ∂O[ω] not containing the cell t ω, and we write*
s*ω∶ S[ω] ↪Ð→ O[ω] for the spine inclusion. The set of spine inclusions is denoted by S.*

* Lemma 3.4.2. For ω_{∈ O, with dim ω ≥ 1 the following square is a pushout and a pullback}*5

*, where all*

*arrows are canonical inclusions:*

*∂O[t ω]* *S[ω]*

*O[t ω]* *∂O[ω].*

**Lemma 3.4.3. Let n**≥ 1, ν ∈ On,*[l] ∈ ν*∣*, and ψ* ∈ O*n*−1 *be such that e _{[l]}ν*

*= t ψ, so that the grafting*

*ν*○* _{[l]}*Y

*ψ*

*is well-defined. Then the following square is a pushout:*

*O*[e* _{[l]}ν*]

*O[ψ]*

*S[ν]* *S[ν ○ _{[l]}*Y

*ψ].*

t

e* _{[l]}* s

_{[l]}*Notation 3.4.4. Let F* _{∶ O Ð→ hom C be a function that maps opetopes to morphisms in some category C,}

and M the set of maps defined by M*∶= {F (ω) ∣ ω ∈ O}. Then for n ∈ N, we define M _{≥n}∶= {F (ω) ∣ ω ∈ O_{≥n}*},
and similarly for M

*, M*

_{>n}*, M*

_{≤n}*, and M*

_{<n}*. For convenience, the latter is abbreviated M*

_{=n}*also let M*

_{n}. If m_{≤ n, we}*m,n*= M

*≥m*∩ M

*≤n*. By convention, M

*≤n*

*= ∅ if n < 0. For example, S*≥2 = {s

*ω∣ ω ∈ O*≥2}, and

S*n,n*+1= S*n*∪ S*n*+1.

**Definition 3.4.5. Write O**∶= {∅ ↪*Ð→ O[ω] ∣ ω ∈ O} for the set of initial inclusions of the representables.*

Let

A*k,n*∶= O*<n−k*∪ S*≥n+1.* (3.4.6)

**Lemma 3.4.7.** *(1) Let X* * _{∈ Psh(O) such that S}n,n*+1

*⊥ X. Then Bn*+1

*⊥ X. In general, every*

*morphism in B _{≥n+1}*

*is an S*

_{≥n}-local isomorphism.*(2) Let X* *∈ Psh(O) such that (Sn,n*+1∪ B*n*+2*) ⊥ X. Then Sn*+2*⊥ X. In particular, if (Sn,n*+1∪B*≥n+2*) ⊥

*X, then S _{≥n}⊥ X.*

*Proof.* *(1) Let ω*_{∈ O}_{n}_{+1}. By 3-for-2, since the composite s_{ω}_{∶ S[ω] ↪}_{Ð→ ∂O[ω] ↪}_{Ð→ O[ω] is in S}_{n,n}_{+1}, it
suffices to show that *(S[ω] ↪Ð→ ∂O[ω]) ⊥ X. Let f ∶ S[ω] Ð→ X be a morphism. The existence*
*of a lift ∂O _{[ω] Ð→ X follows from the existence of a lift O[ω] Ð→ X. Next, given two lifts}*

*g, h* *∶ ∂O[ω] Ð→ X of f, by lemma 3.4.2, it suffices to show that they coincide on O[t ω] to*

*show that they are equal. But since they coincide on S[ω], they coincide on S[t ω], and hence*
*on O _{[t ω] since s}_{t ω}_{⊥ X.}*

*(2) Let ω*∈ O*n*+2*and f* *∶ S[ω] Ð→ X. Then the restriction f∣S[t ω]of f to S[t ω] extends to a unique*

*f*∣*O _{S}_{[t ω]}[t ω]*. We now show that the following square commutes:

*∂O[t ω]* *S[ω]*

*O[t ω]* *X.*

*f*
*f*∣*O _{S}_{[t ω]}[t ω]*

(3.4.8)

5_{Recall that in a topos, the pushout of a monomorphism along any arrow is a monomorphism, and the pushout square}

is a pullback square. This property is sometimes called “adhesivity”, and is a consequence of van Kampen-ness, or descent, for pushouts of monomorphisms.

*Tautologically, we have f*_{∣}_{S}_{[t ω]}_{= f∣}O_{S}_{[t ω]}[t ω]_{∣}_{S}_{[t ω]}, and in particular, f_{∣}_{S}_{[t t ω]}_{= f∣}O_{S}_{[t ω]}[t ω]_{∣}* _{S}_{[t t ω]}*. Since
S

*n*−2

*⊥ X, we have f∣O[t t ω]= f∣*

*O[t ω]*

*S[t ω]*∣*O[t t ω]*. Therefore, square (3.4.8) commutes, and by lemma 3.4.2,

*f extends to a unique f*∣*∂O[ω]∶ ∂O[ω] Ð→ O[ω] such that f∣∂O[ω]*∣*O[t ω]= f∣*
*O[t ω]*

*S[t ω]*. Since B*n⊥ X,*

*f*∣*∂O[ω]* *extends to a unique morphism O _{[ω] Ð→ X.}*

_{□}

**Lemma 3.4.9. Let n*** _{∈ N, and ω ∈ O}n*+2

*. Then the inclusion S[t ω] ↪Ð→ S[ω] is a relative Sn*+1

*-cell*

*complex, i.e. a transfinite composition of pushouts of maps in S _{n}*

_{+1}

*.*

*Proof. We show that the morphism S _{[t ω] ↪}_{Ð→ S[ω] is a (finite) composite of pushouts of elements of}*

S*n*+1*. Let X*0 *= S[t ω]. At stage k ≥ 0, let J be the (necessarily finite) set of inclusions j = (j*1*, j*2) in

Psh(O)→ as on the left

*S[ψj*] *Xk*
*O[ψj*] *S[ω],*
*j*1
s_{ψj}*j*2
∐*j∈JS[ψj*] *Xk*
∐*j∈JO[ψj*] *Xk*+1*.*
⌜
*j*1
s_{ψj}*j*2

*where ψj* ∈ O*n*+1*, such that j*1 *does not factor through Xl* *for any l< k. Define Xk*+1 with the pushout

*on the right. There is a k, bounded by the height of the tree ω*_{∈ tr Z}*n*_{, at which the sequence converges}

*to S[t ω] = X*0↪*Ð→ X*1↪*Ð→ . . . ↪Ð→ Xk= S[ω].* □

**Corollary 3.4.10.** *(1) Let n _{∈ N, and ω ∈ O}_{n}*

_{+2}

*. Then the target embedding t ω*

_{Ð→ ω of ω is an}S*n+1,n+2-local isomorphism.*

*(2) Let k, n _{∈ N, and ω ∈ O}_{≥n+2}. Then the target embedding t ω_{Ð→ ω is an A}_{k,n}-local isomorphism.*

*Proof.*(1) In the square below

*S[t ω]* *O[t ω]*

*S[ω]* *O[ω]*

s*t ω*

*r* t

s*ω*

*the map r is an S _{n}*

_{+1}-local isomorphism by lemma 3.4.9, and the horizontal maps are in S

*. The result follows by 3-for-2.*

_{n}_{+1,n+2}*(2) Let ω*_{∈ O}*m*. Since S*m−1,m+1*⊂ A*k,n* by definition, this follows from the previous point. □
**Corollary 3.4.11. Let ψ**_{∈ O}*n.*

*(1) t t*= s_{[]}t*∶ ψ Ð→ Iψ* *is in Sn*+2*.*

*(2) The morphisms s*_{[]}*, t∶ ψ Ð→ Yψ* *are Sn+1,n+2-local isomorphisms.*

*Proof.* (1) The embedding t t = s_{[]}t*∶ ψ Ð→ Iψ* is precisely the spine inclusion sI*ϕ* of the degenerate

*(n + 2)-opetope Iψ*.

(2) The source embedding s_{[]}* _{∶ ψ Ð→ Y}_{ψ}* is precisely the spine inclusion s

_{Y}

*of the*

_{ψ}*Y*

_{(n + 1)-opetope}*ψ*. The target embedding t

*∶ ψ Ð→ Yψ*is the morphism t∶ t t I

*ψ*Ð→ t I

*ψ*and is the vertical arrow

in the diagram below.

*ψ= S[Iψ*]

Y*ψ*= t I*ψ* I*ψ.*

t

s_{Iψ}

t

The horizontal arrow is an S*n+1,n+2*-local isomorphism by corollary 3.4.10 and the diagonal arrow

**3.5. Extensions.**

*Reminder 3.5.1. Recall that a functor between small categories u* _{∶ A Ð→ B induces a restriction}
*u*∗ *∶ Psh(B) Ð→ Psh(A) that admits both adjoints u*!*⊣ u*∗*⊣ u*∗, given by pointwise left and right Kan

extensions.

*Notation 3.5.2. Let m _{∈ N and n ∈ N ∪ {∞} be such that m ≤ n, and let O}m,n* be the full subcategory of

*O spanned by opetopes ω such that m ≤ dim ω ≤ n. For instance, Om,*∞= O*≥m*.

* Definition 3.5.3 (Truncation). The inclusion ι≥m*∶ O

*m,n*Ð→ O

*≥m*induces a restriction functor(−)

*m,n*∶

Psh(O*≥m*) Ð→ Psh(O*m,n), called truncation, that have both a left adjoint ι≥m*_{!} *and a right adjoint ι≥m*_{∗} .

*Explicitly, for X* _{∈ Psh(O}_{m,n}_{), the presheaf ι}≥m_{!} *X is the “extension by 0”, i.e.* *(ι≥m*_{!} *X*)*m,n* *= X, and*

*(ι≥m*

! *X*)*ψ* *= ∅ for all ψ ∈ O>n. On the other hand, ι≥m*∗ *X is the “canonical extension” of X intro a*

presheaf over_{O}* _{≥m}*: we have

_{(ι}≥m_{∗}

*X*)

*m,n= X, and B>n⊥ ι≥m*

_{∗}

*X, which uniquely determines ι≥m*

_{∗}

*X.*

*Likewise, the inclusion ι≤n* _{∶ O}*m,n* Ð→ O*≤n* induces a restriction functor Psh(O*≤n*) Ð→ Psh(O*m,n*),

also denoted by_{(−)}_{m,n}*and again called truncation, that have both a left adjoint ι≤n*_{!} and a right adjoint

*ι≤n*_{∗} *. Explicitly, for X* _{∈ Psh(O}_{m,n}_{), the presheaf ι}≤n_{!} *X is the “canonical extension” of X intro a presheaf*

overO* _{≤n}*:

*ι≤n*_{!} *X*= colim

*O[ψ]m,n→X*

*O[ψ].*

*On the other hand, ι≤n*_{∗} *X is the “terminal extension” of X in that* *(ι≤n*_{∗} *X*)*m,n* *= X, and (ι≤n*_{∗} *X*)*ψ* is a

*singleton, for all ψ*_{∈ O}* _{<n}*. Note that O

_{<m}_{⊥ ι}≤n_{∗}

*X, and that it is uniquely determined by this property.*

*For n* < ∞, we write (−)* _{≤n}* for (−)

*0,n*∶ Psh(O≥0) = Psh(O) Ð→ Psh(O

*0,n*) = Psh(O

*≤n*), and let

(−)*<n* = (−)*≤n−1* *if n*≥ 0. Similarly, we note (−)*m,n* ∶ Psh(O≤∞) = Psh(O) Ð→ Psh(O*m,*∞) = Psh(O*≥m*)

by _{(−)}* _{≥m}*, and let

_{(−)}

_{>m}_{= (−)}

*.*

_{≥m+1}**Proposition 3.5.4.** *(1) The functors ι≥m*_{!} *, ι≥m*_{∗} *, ι≤n*_{!} *, and ι≤n*_{∗} *are fully faithful.*
*(2) A presheaf X*_{∈ Psh(O}_{≥m}_{) is in the essential image of ι}≥m_{!} *if and only if X _{>n}_{= ∅.}*

*(3) A presheaf X* _{∈ Psh(O}_{≥m}_{) is in the essential image of ι}≥m_{∗} *if and only if for all ω*_{∈ O}_{>n}*we have*

(b*ω*)*≥m⊥ X.*

*(4) A presheaf X*∈ Psh(O_{≤n}) is in the essential image of ι≤n_{∗} *if and only if for all ω*∈ O_{<m}*we have*

(o*ω*)*≤n⊥ X, i.e. Xω* *is a singleton.*

*Proof. The first point follows from the fact that ι≥m* *and ι≤n* are fully faithful, and [SGA72, exposé I,

proposition 5.6]. The rest is straightforward verifications. _{□}

*Notation 3.5.5. To ease notations, we sometimes leave truncations implicit, e.g. point (3) of last *

*propo-sition can be reworded as: a presheaf X* ∈ Psh(O_{≥m}) is in the essential image of ι≥m_{∗} if and only if
B_{>n}⊥ X.

4. The opetopic nerve of opetopic algebras

**4.1. Opetopic algebras. Let k**_{≤ n ∈ N. Recall from notation 3.5.2 that O}_{n}_{−k,n}_{↪}_{Ð→ O is the full}
*subcategory of opetopes of dimension at least n _{−k and at most n. A k-coloured, n-dimensional opetopic}*

*algebra, or _{(k, n)-opetopic algebra, will be an algebraic structure on a presheaf over O}_{n}_{−k,n}*. Specifically,
we describe a monad on the category

_{Psh(O}

*n−k,n), whose algebras are the (k, n)-opetopic algebras. Such*

*an algebra X has “operations” (its cells of dimension n) that can be “composed” in ways encoded by*
*(n + 1)-opetopes*6_{. The operations of X will be “coloured” by its cells dimension}_{< n, which determines}

which operations can be composed together.

As we will see, the fact that the operations and relations of an* _{(k, n)-opetopic algebra are encoded}*
by opetopes of dimension

*of*

_{> n results in the category Alg}_{k,n}

_{(k, n)-opetopic algebras always having a}*canonical full and faithful nerve functor to the category*Psh(O) of opetopic sets (theorem 4.5.11).

We now claim some examples. A classification of*(k, n)-opetopic algebras is given by proposition 4.3.7.*

**Example 4.1.1 (Monoids and categories). Let k***= 0 and n = 1. Then On−k,n* = O1 = {∗}, and

Psh(O*n−k,n) = Set. The category of (0, 1)-opetopic algebras is precisely the category of associative*

*monoids. If k*= 1 instead, then O*n−k,n*= O*0,1*, andPsh(O*n−k,n*) = Graph, the category of directed graphs.

The category of_{(1, 1)-opetopic algebras is precisely the category of small categories.}

**Example 4.1.2 (Coloured and uncoloured planar operads). Let k**_{= 0 and n = 2. Then O}_{n}_{−k,n}_{= O}2≅ N,

and Psh(O2*) ≃ Set/N. The category of (0, 2)-opetopic algebras is precisely the category of planar,*

uncoloured* _{Set-operads. If k = 1 instead, then Psh(O}1,2*) is the category of planar, coloured collections.

The category of_{(1, 2)-opetopic algebras is precisely the category of planar, coloured Set-operads.}

* Example 4.1.3 (Combinads). Let k= 0 and n = 3. Then On−k,n*= O3 is the set of planar finite trees,

and* _{(0, 3)-opetopic algebras are exactly the combinads over the combinatorial pattern of non-symmetric}*
trees, presented in [Lod12].

**4.2. Parametric right adjoint monads. In preparation to the main results of this section, we survey**
elements of the theory of parametric right adjoint (p.r.a.) monads, which will be essential to the definition
and description of * _{(k, n)-opetopic algebras. A comprehensive treatment of this theory can be found in}*
[Web07].

**Definition 4.2.1 (Parametric right adjoint). Let T**_{∶ C Ð→ D be a functor, and let C have a terminal}

*object 1. Then T factors as*

C = C/1 *T*1

*Ð→ D/T 1 Ð→ D.*

*We say that T is a parametric right adjoint (abbreviated p.r.a.) if T*_{1} *has a left adjoint E.*

We immediately restrict ourselves to the case*C = D = Psh(A) for a small category A. Then T*1 is the

*nerve of the restriction E _{∶ A/T 1 Ð→ Psh(A), and the usual formula for nerve functors gives}*

*T Xa*= ∑
*x∈T 1a*

*Psh(A)(Ex, X)* (4.2.2)

*for X _{∈ Psh(A) and a ∈ A. In fact, it is clear that the data of the object T 1 ∈ Psh(A) and of the functor}*

*E completely describe (via equation (4.2.2)) the functor T up to unique isomorphism.*

**Definition 4.2.3 (P.r.a monad). A p.r.a. monad is a monad whose endofunctor is a p.r.a. and whose**

unit and multiplication are cartesian natural transformations7.

*Assume now that T* _{∶ Psh(A) Ð→ Psh(A) is a p.r.a. monad. Define Θ}_{0} to be the full subcategory
of *Psh(A) spanned by the image of E ∶ A/T 1 Ð→ Psh(A). Objects of Θ*0 *are called T -cardinals. By*

[Web07, proposition 4.20], the Yoneda embedding_{A ↪}_{Ð→ Psh(A) factors as}
A↪Ð→ Θ*i* 0

*i*0

↪Ð→ Psh(A)
*or in other words, representable presheaves are T -cardinals.*

*Every p.r.a. monad on a presheaf category is an example of a monad with arities [BMW12]. The*
theory of monads with arities provides a remarkable amount of information about the free-forgetful
adjunctionPsh(A) Ð→* _{←Ð Alg(T ) and about the category of algebras Alg(T ). We summarise those results}*
that we will use below.

**Proposition 4.2.4. The fully faithful functor i**_{0} _{∶ Θ}_{0}_{↪}_{Ð→ Psh(A) is dense, or equivalently, its associated}

*nerve functor N*0∶ Psh(A) ↪Ð→ Psh(Θ0*) is fully faithful.*

7_{P.r.a. monads on presheaf categories are a strict generalisation of polynomial monads, since an endofunctor}_{Set/I Ð→}

*Proof. We denote the inclusion* A ↪Ð→ Θ0 *by i, and note that N*0 *is isomorphic to i*∗ ∶ Psh(A) Ð→

Psh(Θ0*). Now, i is fully faithful, and by [SGA72, exposé I, proposition 5.6], this is equivalent to i*∗

being fully faithful. _{□}

**Corollary 4.2.5. Let J**_{A}_{∶={ε}_{θ}_{∶ i}_{!}*i*∗*θÐ→ θ ∣ θ ∈ Θ*0*− im i}, where εθ* *is the counit at θ. Then a presheaf*

*X*∈ Psh(Θ0*) is in the essential image of N*0 *if and only if J*_{A}*⊥ X.*

*Proof. By the formula i*_{∗}*Y* = Psh(A)(Θ0*(i−, −), Y ) we have the sequence of isomorphisms*

*(i*∗*Y*)*θ*= Psh(A)(Θ0*(i−, θ), Y )*

*≅ Psh(A)(i*∗*θ, i*∗*i*_{∗}*Y*) *since i*_{∗} is fully faithful
≅ Psh(Θ0*)(i*!*i*∗*θ, i*∗*Y*) *since ι*!*⊣ i*∗*.*

*It is easy to check that one direction of the previous isomorphism is pre-composition by εθ*. Conversely,

*take X*_{∈ Psh(Θ}0*) such that εθ⊥ X for all θ ∈ Θ*0. Then

*i*∗*i*_{∗}*Xθ*≅ Psh(Θ0*)(θ, i*∗*i*∗*X*)

≅ Psh(Θ0*)(i*!*i*∗*θ, X*)

≅ Psh(Θ0*)(θ, X)* *since εθ⊥ X*

*≅ Xθ,*

*and thus X* *∈ im i*∗*. Thus a presheaf X is isomorphic to one of the form i*_{∗}*Y if and only if we have*
*εθ* *⊥ X for all θ ∈ Θ*0*. But if θ∈ im i, then the associated counit map is already an isomorphism, hence*

*we can restrict to J*_{A}. □

*Notation 4.2.6. Let the “identity-on-objects / fully faithful” factorisation of the composite functor*
*FTi*0∶ Θ0 ↪*Ð→ Psh(A) Ð→ Alg(T ) be denoted*

Θ0

*t*

Ð→ Θ*T*
*iT*

↪*Ð→ Alg(T )* (4.2.7)

**Theorem 4.2.8.** *(1) The fully faithful functor i _{T}*

_{∶ Θ}

_{T}_{↪}

_{Ð→ Alg(T ) is dense, or equivalently, its}*associated nerve functor NT*

*∶ Alg(T ) ↪*Ð→ Psh(Θ

*T) is fully faithful.*

*(2) The following diagram is an exact adjoint square*8*.*

Psh(A) *Alg(T )*
Psh(Θ0) Psh(Θ*T*)
*N*0
*FT*
*NT*
*UT*
⊥
*t*!
*t*∗
⊥

*In particular, both squares commute up to natural isomorphism.*

*(3) (Nerve theorem) Any X* _{∈ Psh(Θ}_{T}_{) is in the essential image of N}_{T}*if and only if t*∗*X is in the*
*essential image of N*0*.*

*Proof. See [Web07, theorem 4.10], and [BMW12, proposition 1.9].* _{□}

**Corollary 4.2.9. Let J**_{T}_{∶= t}_{!}*J*_{A}*= {t*!*εθ∶ t*!*i*!*i*∗*θÐ→ t*!*θ∣ θ ∈ Θ*0*− im i}, where εθ∶ i*!*i*∗*θÐ→ θ is the counit*

*at θ. Then X*∈ Psh(Θ*T) is in the essential image of NT* *if and only if JT* *⊥ X.*

*Proof. This follows from theorem 4.2.8 point (3), and from corollary 4.2.5.* □

8_{There exists a natural isomorphism N}

0*UT≅ t*∗*NT* *whose mate t*!*N*0*Ð→ NTFT*is invertible (satisfies the Beck-Chevalley

**4.3. Coloured Z***n*_{-algebras. In this section, we extend the polynomial monad Z}*n* _{over} _{Set/O}
*n* =

Psh(O*n*) to a p.r.a. monad Z*n* overPsh(O*n−k,n), for k ≤ n ∈ N.*

This new setup will encompass more known examples (see proposition 4.3.7). For instance, recall that
the polynomial monad Z2 _{on} _{Set/O}

2≅ Set/N is exactly the monad of planar operads. The extension of

Z2 *will retrieve coloured planar operads as algebras. Similarly, the polynomial monad Z*1 _{on} _{Set is the}

free-monoid monad, which we would like to vary to obtain “coloured monoids”, i.e. small categories.
*Let k _{≤ n ∈ N. Let us define a p.r.a. endofunctor Z}n*

_{on the category}

_{Psh(O}

*n−k,n*), that will restrict to

the polynomial monad Z*n*_{∶ Psh(O}

*n*) Ð→ Psh(O*n) in the case k = 0. Following section 4.2, to define the*

p.r.a. endofunctor Z*n* _{as the composite}

Psh(O*n−k,n*)

Z*n*1

Ð→ Psh(O*n−k,n*)/Z*n*1≃ Psh(O*n−k,n*/Z*n*1) Ð→ Psh(O*n−k,n),*

up to unique isomorphism, it suffices to define its value Z*n*_{1 on the terminal presheaf, and to define a}

*functor E*∶ O*n−k,n*/Z*n*1Ð→ Psh(O*n−k,n*).
**Definition 4.3.1. Define Z***n*_{1 as:}

(Z*n*_{1}_{)}

*ψ∶={∗},* (Z*n*1)*ω∶={ν ∈ On*+1*∣ t ν = ω}.*

*where ψ*_{∈ O}_{n}_{−k,n−1}*and ω*_{∈ O}_{n}. We define the functor E_{∶ O}_{n}_{−k,n}_{/Z}*n*_{1}_{Ð→ Psh(O}

*n−k,n*) as follows. On

objects, for∗ ∈ (Z*n*_{1)}

*ψ* *and ν*∈ (Z*n*1)*ω*, let

*E(∗) ∶= O[ψ],* *E(ν) ∶= S[ν],*

and on morphisms as the canonical inclusions. The functor Z*n*

1 ∶ Psh(O*n−k,n*) Ð→ Psh(O*n−k,n*/Z*n*1) is

*defined as the right adjoint to the left Kan extension of E along the Yoneda embedding. We now recover*
the endofunctor Z*n* _{explicitly using equation (4.2.2): for ψ}_{∈ O}

*n−k,n−1* we have (Z*nX*)*ψ* *≅ Xψ*, and for

*ω*∈ O*n* we have
(Z*n _{X}*

_{)}

*ω*≅ ∑

*ν*∈O

*n*+1

*t ν=ω*Psh(O

*n−k,n)(S[ν], X).*Note that

_{(Z}

*n*

_{X}_{)}

*ω* matches with the “uncolored” version of Z*n* of equation (3.1.2).

*Recall from section 4.2 that a p.r.a. monad is a monad M whose unit id* *Ð→ M and *
*multiplica-tion M M* * _{Ð→ M are cartesian, and such that M is a p.r.a. endofunctor. We now endow the p.r.a.}*
endofunctor Z

*n*

_{with the structure of a p.r.a. monad over}

_{Psh(O}

*n−k,n*). We first specify the unit and

*multiplication η*1 ∶ 1 Ð→ Z*n1 and µ*1 ∶ Z*n*Z*n*1 Ð→ Z*n*1 on the terminal object 1, and extend them to

cartesian natural transformations (lemma 4.3.3). Next, we check that the required monad identities
hold for 1 (lemma 4.3.4), which automatically gives us the desired monad structure on Z*n*_{.}

**Lemma 4.3.2. The polynomial functor Z**n_{Z}*n*_{∶ Set/O}

*n*Ð→ Set/O*n* *is given by*

O*n* *E* O(2)*n*+2 O*n,*

e t t

*where*

*(1)* _{O}(2)_{n}_{+2} *is the set of* _{(n + 2) opetopes of height 2, i.e. of the form}

Y*ν* ◯

*[[pi*]]

Y*νi,*

*with ν, ν _{i}*

_{∈ O}

_{n}_{+1}

*and*

*●*

_{[p}_{i}_{] ranging over a (possibly empty) subset of ν}*,*

*(2) for ξ*

_{∈ O}(2)

_{n}_{+2}

*, E*∣

_{(ξ) = ξ}*,*

*(3) for ξ*_{∈ O}(2)_{n}_{+2} *and* * _{[l] ∈ E(ξ) = ξ}*∣

*, e*

_{[l] = e}_{[l]}t.*We now define η*1

*and µ*1.